Three-Mass Coupled Oscillation Technique for Mechanically Robust Micromachined Gyroscopes

ABSTRACT

A micromachined gyroscope is disclosed comprising a substrate, three masses m 1 , m 2 , and m 3 , configured to oscillate along a first direction x or y, whereby the first mass m 1  is mechanically coupled to the substrate, the second mass m 2  is mechanically coupled to the first mass m 1  and to substrate, and the third mass m 3  is mechanically coupled to the second mass m 2 , whereby the weight and the spring constants k 1 , k 2 , k 3  of the respective masses m 1 , m 2 , and m 3  and mechanical couplings k 12 , k 23  are selected, such that, during operation mass m 2  oscillates at a frequency substantially above the resonance frequencies of mass m 1  and mass m 3 . The resonance frequency of mass m 2  may be at least 2 times, or even 2.5 times, higher than the resonance frequency of mass m 1  or m 3 .

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims priority to U.S. Provisional Patent ApplicationNo. 61/534,146 filed on Sep. 13, 2011, the contents of which are herebyincorporated by reference.

BACKGROUND

Micromachined gyroscopes are angular rate sensors that typically operateaccording to a physical phenomenon called the Coriolis Effect. TheCoriolis Effect is, simply, the deflection of moving objects viewed froma rotating frame. For an object mounted to a substrate, the object tendsto oscillate (e.g., vibrate, move, or drive) in a perpendicular planewhen the substrate rotates. Hence, in order to make use of the CoriolisEffect, micromachined gyroscopes may be composed of an oscillating partcomprising at least one mass, and a sensing part which is free to movein a perpendicular plane of the oscillating part. The sensing part isaffected by the rotation of the gyroscope, as the oscillating part willbe deflected. Under an external rotation, the oscillating mass deflects,and that deflection is sensed via the movement of the sensing part.

The sensitivity of such an oscillating gyroscope depends on itsoscillation magnitude. In order to achieve a stable and largesensitivity, stable and large oscillation amplitude is desirable.

Typically, a large oscillation is achieved by using aone-degree-of-freedom (1-DOF) oscillator that is operated at itsresonant frequency. Stability is then obtained with the help ofstabilization circuitry (e.g., phase lock loops (PLLs), proportionalintegral (PI) controllers, etc.) to keep the gyroscope operating nearthis resonance frequency.

In some cases, the 1-DOF oscillator may be operated at non-resonancefrequency, thereby reducing the need for stabilisation circuitry.However, a magnitude of the oscillation at non-resonance frequencieswill be less than a magnitude of the oscillation at the resonancefrequency. When the oscillator is oscillating at non-resonancefrequencies, though, changes in the frequency, as well as the qualityfactor, will have a lesser effect on the oscillation magnitude, ascompared to when the oscillator is oscillating at the resonancefrequency.

Typical gyroscopes consume 10 to 20 times more power than a typicalaccelerometer in commercial applications. Some of this power consumptionresults from the comb-drive actuation used in typical gyroscopes toobtain large oscillation magnitudes. Comb-drive actuation involveselectrostatic forces being generated between two comb-like structures.One comb is fixed to the substrate while the other comb is movable. Theforce developed by the comb-drive actuator is proportional to the changein capacitance between the two combs. However, this capacitanceincreases with driving voltage difference between both combs, with thecoupling area reflected by the number of comb teeth, and the gap betweenthese teeth. As a result, achieving large oscillation magnitudes withcomb-drive actuation requires large polarization voltage differences,typically 12V in commercial devices. Such high polarization voltagedifferences are not conducive to a low-power gyroscope. Another sourceof this power consumption may be stabilization circuitry, such as PLLsand/or PI controllers, used to stabilize the oscillation increase powerconsumption of the gyroscope, which is similarly not conducive to alow-power gyroscope. Other sources of power consumption exist as well.

One option for reducing the power consumption of a gyroscope is to use atwo-degree-of-freedom (2-DOF) oscillator that includes two masses and,accordingly, has two resonance frequencies. The 2-DOF gyroscope may beoperated in between the two resonance frequencies. The amplituderesponse typically has minimal dependency on the varying quality factorand the resonance frequencies. However, the magnitude of this responseis still very small and comparable to the non-resonance response of the1-DOF oscillator discussed above.

Accordingly, a micromachined gyroscope with reduced power consumptionmay be desirable. It may be desirable for such a micromachined gyroscopeto have a stable oscillation frequency range with a high mechanicalamplification between the actuator and the driving part.

SUMMARY

Disclosed is a gyroscope with reduced power consumption, as compared totypical gyroscopes. The disclosed gyroscope is designed without the needof stabilization circuitry, and with a reduced need for driving andcontrolling circuitry, thereby reducing the power consumption of thegyroscope.

In one aspect, a micromachined gyroscope is disclosed. The micromachinedgyroscope comprises a substrate and at least three masses (m₁, m₂, m₃).The first mass m₁ is mechanically coupled to the substrate via amechanical connection k₁, the second mass m₂ is mechanically coupled tothe first mass m₁ via a connection k₁₂ and to the substrate via amechanical connection k₂, and the third mass m₃ is mechanically coupledto the second mass m₂ via a mechanical connection k₂₃. The three massesare each configured to oscillate along a first direction x or y.

The following relationships exist between the masses m₁, m₂, m₃, and themechanical connections k₁, k₂, k₁₂, k₂₃:

[(k ₂ +k ₁₂ +k ₂₃)/m ₂]>>([(k ₁ +k ₁₂)/m ₁]˜[(k ₂₃)/m ₃])

In some embodiments, the third mass m₃ is also mechanically coupled tothe substrate via a mechanically connection k₃.

The masses m₁, m₂, and m₃ can be the driving masses of the gyroscopeconfigured to oscillate along a first direction x. To this end, thegyroscope may further comprise actuators for stimulating these drivingmasses. These actuators may be parallel plate actuators.

In another embodiment, the micromachined gyroscope further comprises aduplicate m₁′, m₂′, and m₃′ of these 3 mass configuration and thisduplicate is configured to oscillate along the first direction x but inopposite phase with these three masses m₁, m₂, and m₃.

The masses m₁, m₂, and m₃ can be the sensing masses of the gyroscope,which are configured to oscillate along a first direction y when thegyroscope is rotating.

In some embodiments, the three masses m₁, m₂, m₃ can be configured tooscillate in a linear way.

In another aspect, a micromachined gyroscope is disclosed comprising asubstrate, a driving mass mechanically coupled to a sensing mass, bothmasses being movable in perpendicular directions and, when in operationunder the influence of the Coriolis force, the driving mass causes thedriving of the sensing mass, whereby at least one of the driving mass orof the sensing mass is configured as a connection of three masses m₁,m₂, m₃, whereby the first mass m₁ is mechanically coupled to thesubstrate, the second mass m₂ is mechanically coupled to the first massm₁ and to the substrate, and the third mass m₃ is mechanically coupledto the second mass m₂, whereby the following relationship exist:

[(k ₂ +k ₁₂ +k ₂₃)/m ₂]>>([(k ₁ +k ₁₂)/m ₁]˜[(k ₂₃)/m ₃])

with m₁, m₂, m₃ being the weight of respectively mass m₁, m₂ and m₃,with k₁, k₂ (and k₃) being the spring constant of the mechanicalconnection between of the respective mass m₁ or m₂ or m₃ and thesubstrate, and with k₁₂, k₂₃ being the spring constant of the mechanicalconnection between mass m₂ and mass m₁ or mass m₃ respectively.

In yet another aspect, a micromachined gyroscope is disclosed comprisinga substrate, three masses m₁, m₂, and m₃ , configured to oscillate alonga first direction x or y, whereby the first mass m₁ is mechanicallycoupled to the substrate, the second mass m₂ is mechanically coupled tothe first mass m₁ and to substrate, and the third mass m₃ ismechanically coupled to the second mass m₂, whereby the weight and thespring constants k₁, k₂, k₃ of the respective masses m₁, m₂ and m₃ andmechanical couplings k₁₂, k₂₃ are selected, such that, during operationmass m₂ oscillates at a frequency substantially above the resonancefrequencies of mass m₁ and mass m₃.

In some embodiments, the resonance frequency of mass m₂ is at least 2times, or even 2.5 times, higher than the resonance frequency of mass m₁or m₃.

In yet another aspect, a method for designing a micromachined gyroscopeis disclosed. This micromachined gyroscope comprises a substrate, atleast 3 masses m₁, m₂ and m₃ being configured to oscillate along a firstdirection x or y, whereby the first mass m₁ is mechanically coupled tothe substrate, the second mass m₂ is mechanically coupled to the firstmass m₁ and to substrate, and the third mass m₃ is mechanically coupledto the second mass m₂, whereby the following relationship exist:

[(k ₂ +k ₁₂ +k ₂₃)/m ₂]>>([(k ₁ +k ₁₂)/m ₁]˜[(k ₂₃)/m ₃])

with m₁, m₂, m₃ being the weight of respectively mass m₁, m₂ and m₃,with k₁, k₂ (and k₃) being the spring constant of the mechanicalconnection between of the respective mass m₁ or m₂ or m₃ and thesubstrate, and with k₁₂, k₂₃ being the spring constant of the mechanicalconnection between mass m₂ and mass m₁ or mass m₃ respectively.

The method comprises: selecting m₁, m₃, k₁ and k₃ whereby[(k₁+k₁₂)/m₁]˜[(k₃+k₂₃)/m₃], and selecting m₂, k₂ such that duringoperation of the gyroscope:

[(k ₂ +k ₁₂ +k ₂₃)/m ₂]>>([(k ₁ +k ₁₂)/m ₁]>[(k ₃ +k ₂₃)/m ₃]).

The method further comprises selecting a mechanical amplificationbetween the movement of mass m₁ and mass m₃ and dimensioning k₂ in viewof this desired mechanical amplification.

BRIEF DESCRIPTION OF THE DRAWINGS

Exemplary embodiments are illustrated in referenced figures of thedrawings. It is intended that the embodiments and figures disclosedherein are to be considered illustrative rather than restrictive.

FIG. 1 shows a schematic mechanical equivalent of a micromachinedgyroscope comprising three driving masses m₁, m₂, m₃ and one sensingmass m_(sense), in accordance with an embodiment.

FIG. 2 shows a schematic mechanical equivalent of a micromachinedgyroscope comprising three driving masses m₁, m₂, m₃ and one sensingmass m_(sense), in which mass m₃ is coupled to the substrate, inaccordance with an embodiment.

FIG. 3 shows the resonance behavior of the gyroscope shown in FIG. 2:normalized displacement response (unitless) vs. frequency (Hz), inaccordance with an embodiment.

FIG. 4 shows a schematic mechanical equivalent of a micromachinedgyroscope comprising three driving masses m₁, m₂, m₃ and one sensingmass m_(sense) whereby the sensing mass is decoupled from the drivingmass m₃, in accordance with an embodiment.

FIG. 5 shows a schematic mechanical equivalent of a micromachinedgyroscope comprising three driving masses m₁, m₂, m₃ and 3 sensing massm_(sense-2), m_(sense-3) whereby the sensing masses are decoupled fromthe driving mass m₃, in accordance with an embodiment.

FIG. 6 shows a schematic mechanical equivalent of a micromachinedgyroscope comprising the driving masses m₁, m₂, m₃ are arranged in atuning fork configuration and one sensing mass m_(sense), in accordancewith an embodiment.

DETAILED DESCRIPTION

The present disclosure contains particular embodiments and withreference to certain drawings but the disclosure is not limited thereto.The drawings described are only schematic and are non-limiting. In thedrawings, the size of some of the elements may be exaggerated and notdrawn on scale for illustrative purposes. The dimensions and therelative dimensions do not correspond to actual reductions to practiceof the disclosure. Reference throughout this specification to “oneembodiment” or “an embodiment” means that a particular feature,structure or characteristic described in connection with the embodimentis included in at least one embodiment of the present disclosure. Thus,appearances of the phrases “in one embodiment” or “in an embodiment” invarious places throughout this specification are not necessarily allreferring to the same embodiment, but may refer to differentembodiments. Furthermore, the particular features, structures orcharacteristics may be combined in any suitable manner, as would beapparent to one of ordinary skill in the art from this disclosure, inone or more embodiments. The terms top, bottom, over, under and the likein the description and the claims are used for descriptive purposes andnot necessarily for describing relative positions. It is to beunderstood that the terms so used are interchangeable under appropriatecircumstances and that the embodiments of the disclosure describedherein are capable of operation in other orientations than described orillustrated herein. It is to be noticed that the term “comprising”, usedin the claims, should not be interpreted as being restricted to themeans listed thereafter; it does not exclude other elements or steps. Itis thus to be interpreted as specifying the presence of the statedfeatures, integers, steps or components as referred to, but does notpreclude the presence or addition of one or more other features,integers, steps or components, or groups thereof. Thus, the scope of theexpression “a solution comprising components A and B” should not belimited to solution consisting only of components A and B. It means thatwith respect to the present disclosure, the only relevant components ofthe solution are A and B.

In this disclosure a micromachined gyroscope is disclosed. Such amicromachined gyroscope is an angular rate sensor that operatesaccording to the Coriolis Effect described above. Such a micromachinedgyroscope is manufactured using semiconductor process manufacturingsteps.

More particularly, a micromachined gyroscope is disclosed comprising aconfiguration of 3 masses mechanically coupled to oscillate along afirst direction y. In such a three-mass oscillation scheme, illustratedby FIG. 1, mass m₁ is coupled to mass m₂ and to the substrate, mass m₂is coupled to mass m₃ and to the substrate, while mass m₃ drives thesensing part m_(sense). Mass m₃ is only mechanically coupled to mass m₁via the second mass m₂.

Mass m₁ is driven by actuators, which are typically electrostaticallyactuated. Although comb-drive actuators, when operated at lowervoltages, can be used, it may be desirable to use parallel plateactuators, as they operate at lower voltages. Such parallel-plateactuators are more power-efficient, although they cannot provide largedisplacements due to their non-linear behavior. However, thanks to themechanical amplification between the movement of mass m₁ and mass m₃ asdiscussed below, the small displacement of the parallel-plate actuatorscan be amplified, resulting in an appropriately higher oscillationamplitude of the third mass m₃. For example, the amplitude increase frompeak to peak may be several micrometers.

The third mass m₃ is used as the oscillating mass which creates theCoriolis force upon external rotation. The deflection of mass m₃ issensed by the mass m_(sense) moving in a direction y perpendicular tothe direction x along which the driving masses m₁, m₂, and m₃oscillates. This mass m_(sense) is, in the configuration illustrated byFIG. 1, directly coupled to mass m_(3drive) and as such is part of thedriving mass m₃.

The values of the masses m₁, m₂, m₃, spring constants k₁, k₂, k₃, k₁₂,k₂₃, and the damping levels b₁ and b₂ are designed to result in a large,and flat (e.g., constant) over a frequency range (e.g., 50 Hz or above),displacement response for mass m₃, and a mechanical amplificationbetween mass m₁ and mass m₃ at this flat frequency response of mass m₃.Hence, the amplitude of the oscillation of mass m₁ can be small,typically less than 200 nm, or even 100 nm. As discussed above, thissmall oscillation amplitude of mass m₁ allows low voltage actuation ofactuators (e.g., parallel-plate actuators or comb-drive actuators).Hence, the response of mass m₃ to the actuators will be robust withoutneeding of any external circuitry, and overall power consumption willdrop.

Whereas in FIG. 1 mass m₃ was not coupled to the substrate, FIG. 2illustrates another embodiment where mass m₃ is mechanically coupled tothe substrate as well. This coupling is modeled by a spring k₃ and adamping b₃. Such a configuration would accommodate for the imperfectionscoming from the fabrication of the micromachined gyroscope. If thesprings k₁ and k₃ are designed in a similar shape, then even, if thereis a process related imperfection, all springs k₁, k₂, k₃, k₁₂, k₂₃ areaffected to the same degree. Moreover, anchoring all masses m₁, m₂ andm₃ to the substrate minimizes the mechanical stress related tobuckling/bending of the cantilevered masses and allows a larger andflatter device.

The mechanical system illustrated by FIG. 2 can be modeled by threeequations (1):

${Force} = {{m_{1}\frac{\partial^{2}x_{1}}{\partial x^{2}}} + {b_{1}\frac{\partial x_{1}}{\partial x}} + {k_{1}x_{1}} + {\left( {x_{1} - x_{2}} \right)k_{12}}}$$0 = {{m_{2}\frac{\partial^{2}x_{2}}{\partial x^{2}}} + {b_{2}\frac{\partial x_{2}}{\partial x}} + {k_{2}x_{2}} + {\left( {x_{2} - x_{1}} \right)k_{12}} + {\left( {x_{2} - x_{3}} \right)k_{23}}}$$0 = {{m_{3}\frac{\partial^{2}x_{3}}{\partial x^{2}}} + {b_{3}\frac{\partial x_{3}}{\partial x}} + {k_{3}x_{3}} + {\left( {x_{3} - x_{2}} \right)k_{23}}}$

From these 3 equations, the displacement responses of each mass m₁, m₂and m₃ can be derived analytically (2):

$X_{1} = \frac{Force}{k_{1} + k_{12} + {{j\omega}\; b_{1}} - {m_{1}\omega^{2}} - \frac{k_{12^{2}}}{\begin{matrix}{k_{12} + k_{2} + k_{23} + {{j\omega}\; b_{2}} - {m_{2}\omega^{2}} -} \\\frac{k_{23}^{2}}{k_{23} + k_{3} + {{j\omega}\; b_{3}} - {m_{3}\omega^{2}}}\end{matrix}}}$$\mspace{20mu} {X_{2} = \frac{X_{1}k_{12}}{k_{12} + k_{2} + k_{23} + {{j\omega}\; b_{2}} - {m_{2}\omega^{2}} - \frac{k_{23}^{2}}{k_{23} + k_{3} + {{j\omega}\; b_{3}} - {m_{3}\omega^{2}}}}}$$\mspace{20mu} {X_{3} = \frac{X_{2}k_{23}}{k_{23} + k_{3} + {{j\omega}\; b_{3}} - {m_{3}\omega^{2}}}}$

FIG. 3 illustrates the resonance behavior of the three mass coupledoscillation configuration. The frequency of the actuating force is sweptand the displacement of the actuated mass is measured with respect tothe DC displacement. Hence, the y-axis of FIG. 3 is representative forthe quality factor of the oscillation peaks. In-between response peaksf₁ and f₃ a substantially flat response region is obtained. These tworesonance peaks f₁ and f₃ are determined by the mass m₁ and mass m₃,when the resonance frequency of mass m₂ is selected to be higher thaneither of both resonance peaks.

In order to achieve a flat and large mass m₃ response and a mechanicalamplification between mass m₁ and mass m₃, the following design methodis applied. First, mass m₂ is considered to be a non-moving rigid bodywhereby k₂ is assumed to be infinitive. Hence, the motion of mass m₁ andmass m₃ can be determined separately. Then, the resonant frequencies f₁and f₃ of respectively mass m₁ and mass m₃ are equated to each other,assuming that mass m₂ had no impact, as shown in equation (3):

$\frac{k_{1} + k_{12}}{m_{1}} = \frac{k_{23} + k_{3}}{m_{3}}$

If the damping levels b₁ and b₂, and when coupled to the substrate b₃,are low enough, the finite k₂ value will cause the resonant frequenciesf₁ of mass m₁ and f₃ of mass m₃ to separate from each other and form arobust response plateau in-between and a mechanical amplificationbetween mass m₁ and mass m₃. The separation of mass m₁ and mass m₃resonant frequencies and the response level of mass m₃ at the plateaudepend on the value of k₂. The higher k₂ is, the smaller the separationand the larger the response will be.

This mechanical amplification can be further improved by increasing thevacuum level of the environment in which the gyroscope operates or thequality factor of the individual peaks.

The position of the anti-resonance frequency of mass m₁, where themechanical amplification is the highest from mass m₁ towards mass m₃,can be tuned by changing k₁. The value of k₁ can be easily tuned ifparallel plate actuators are used to actuate mass m₁. However, from theapplication point of view, one might prefer to not operate at theanti-resonance frequency of mass m₁, because that will be unstable formass m₁. In that case, the gyroscope is operated slightly off theanti-resonance frequency of mass m₁ and the mechanical amplificationratio from mass m₁ to mass m₃ will be around 20-30 regardless of thevacuum level.

The damping levels b₁, b₂ (and b₃ when present) or the quality factorsof each resonant peak f₁ and f₃ have an important role on the operationof the gyroscope. If the quality factors are not large enough, thecoupling cannot occur and the plateau cannot be formed.

The selection of the quality factors of the resonance frequencies f₁ andf₃ is a design criterion. The larger k₂ is, the larger the qualityfactors of both resonance frequencies should be. As a rule of thumb,these quality factor values should be one order of magnitude larger thanthe ratio of mass m₂ resonant frequency to the average frequency of theplateau between the resonance frequency f₁ and f₃ of mass m₁ and m₃ whencoupled via mass m₂ due to the finite value of spring k₂.

The position of the anti-resonance of mass m₁ does not have to be at themid-point of this plateau. This position depends on the ratio of mass m₁to mass m₃, but can be tuned by altering k₁. So, although during theinitial design phase the resonant frequencies of m₁ and m₃ are equatedto each other f₁˜f₃, thereby assuming mass m₂ to be a non-moving body,at the end, equation (3) does not have to hold due to changed k₁.

The mechanical amplification between m₁ and m₃ depends on the operatingfrequency. If the operating frequency is at the anti-resonance of massm₁, the amplification ratio will be the maximum. However, this situationcan bring instability to the mass m₁ motion. It is proposed to operateslightly off anti-resonance. In this case the mechanical amplificationratio can realistically be 20-30.

FIGS. 4, 5, and 6 show alternative embodiments. The proposed three-masscoupled oscillation technique can be used wherever a 1-DOF oscillator isused within vibrating gyroscopes. Decoupled or non-decoupled sense anddrive schemes and a tuning fork topology can be used. Moreover, thisthree-mass oscillation topology can be used in the sensing part of thegyroscope to achieve a large bandwidth and an amplified sensitivity.

FIG. 4 illustrates a micromachined gyroscope comprising three drivingmasses m₁, m₂, and m₃ with mass m₁ is coupled (k₁, b₁) to the substrateand to (k₁₂) mass m_(z), m₂ is coupled to the substrate (k₂,b₂) and to(k₂₃), while the mass m₃ and mass m_(3drive) drives the sensing massm_(sense) via the decoupling mass m_(decoupling).

FIG. 5 illustrates a micromachined gyroscope comprising three drivingmasses m₁, m₂, and m₃ with mass m₁ being coupled (k₁, b₁) to thesubstrate and to (k₁₂) mass m₂, m₂ is coupled to the substrate (k₂, b₂)and to (k₂₃) the mass m₃ and mass m₃ drives the sensing mass m_(sense)via the decoupling mass m_(decoupling). In this embodiment also thesensing mass m_(sense) is configured as a connection for thee massesm_(sense 1), m_(sense 2), m_(sense 3), whereby mass m_(sense) 2 iscoupled to the substrate and to mass m_(sense 2), m_(sense 2) is coupledto the substrate and the mass m_(sense) 3. In this configuration astable oscillation frequency range for the driving masses is obtainedwhereby the movement of mass m₁ is mechanically amplified to mass m₃,but also a stable sensing frequency range is obtained whereby themovement of mass m_(sense 1) is mechanically amplified to massm_(sense 3).

FIG. 6 illustrates a micromachined gyroscope comprising three drivingmasses m₁, m₂, and m₃ in a tuning fork configuration with mass m₁ beingcoupled (k₁, b₁) to the substrate and to (k₁₂) mass m₂, mass m₂ beingcoupled to the substrate (k₂,b₂) and to (k₂₃) the mass m₃, and massm_(3drive) driving the sensing mass m_(sense). It further comprises asecond series of three driving masses m₁′, m₂′, and m₃′ with mass m₁′being coupled to mass m₁ and to (k₁₂′) mass m₂′, m₂′ being coupled tothe substrate and to (k₂₃′) the mass m₃′, and mass m₃′ driving thesensing mass ms_(sense)′. Both series of three mechanically coupleddriving mass m₁, m₂, and m₃ and m₁′, m₂′, and m₃′ are actuated by thesame actuators.

A number of example embodiments are contemplated. In one exampleembodiment, a micromachined gyroscope may include a substrate and threemasses configured to oscillate along a first direction. The first massm₁ may be mechanically coupled to the substrate, the second mass m₂ maybe mechanically coupled to the first mass m₁ and to substrate, and thethird mass m₃ may be mechanically coupled to the second mass m₂. Thegyroscope may be defined as follows:[(k₂+k₁₂+k₂₃)/m₂]>>([(k₁+k₁₂)/m₁]˜[(k₂₃)/m₃]), where m₁, m₂, m₃ are theweights of, respectively, the masses m₁, m₂ and m₃, k₁, k₂ being thespring constant of the mechanical connection between of the respectivemass and the substrate, and k₁₂, k₂₃ being the spring constant of themechanical connection between m₂ and m₁ and m₂ and m₃, respectively.

In some embodiments, the third mass m₃ may be mechanically coupled tothe substrate and the following relationship may exist:

[(k ₂ +k ₁₂ +k ₂₃)/m ₂]>>[(k ₁ +k ₁₂)/m ₁ ]˜[k ₃ +k ₂₃)/m ₃]

where k₃ is the spring constant of the mechanical connection between ofthe mass m₃ and the substrate.

In some embodiments, the three masses may be driving masses. Thegyroscope may further include driving means to drive the first mass m₁.The driving means may be, for example, one or more parallel plateelectrostatic actuators.

In some embodiments, the three masses may be sensing masses configuredto move when the gyroscope rotates.

In some embodiments, the gyroscope may further include a duplicate ofthe three-mass configuration. The duplicate may be configured tooscillate along the first direction in opposite phase with thethree-mass configuration.

In some embodiments, the masses may be configured to oscillate linearly.

In another example embodiment, a method for designing a micromachinedgyroscope may include selecting m₁, m₃, k₁ and k₃ such that[(k₁+k₁₂)/m₁]˜[k₃+k₂₃)/m₃]. The method may further include selecting m₂,k₂ such that during operation:

[(k ₂ +k ₁₂ +k ₂₃)/m ₂]>>[(k ₁ +k ₁₂)/m ₁ ]>[k ₃ +k ₂₃)/m ₃].

In some embodiments, the method may further comprises selecting amechanical amplification between the movement of mass m₁ and mass m₃,and dimensioning k₂ in view of this desired mechanical amplification.

1. A micromachined gyroscope comprising a three-mass configuration thatincludes a substrate and three masses configured to oscillate in a firstdirection, wherein: a first mass having a first weight m₁ ismechanically coupled to the substrate; a second mass having a secondweight m₂ is mechanically coupled to the first mass and to thesubstrate; and a third mass having a third weight m₃ is mechanicallycoupled to the second mass, wherein the following relationship exist:[(k ₂ +k ₁₂ +k ₂₃)/m ₂>>([(k ₁+k₁₂)/m ₁]˜[(k ₂₃)/m _(3])) wherein k₁ isa spring constant of a mechanical connection between the first mass andthe substrate, k₂ is a spring constant of a mechanical connectionbetween the second mass and the third substrate, k₁₂ is a springconstant of a mechanical connection between the first mass and thesecond mass, and k₂₃ is a spring constant of a mechanical connectionbetween the second mass and the third mass.
 2. The micromachinedgyroscope of claim 1, wherein the third mass is mechanically coupled tothe substrate, and wherein the following relationship exists:[(k ₂ +k ₁₂ +k ₂₃)/m ₂]>>[(k ₁ +k ₁₂)/m ₁ ]˜[k ₃ +k ₂₃)/m ₃] wherein k₃is a spring constant of a mechanical connection between the third massand the substrate.
 3. The micromachined gyroscope of claim 1, whereinthe three masses are driving masses, and further comprising drivingmeans to drive the first mass.
 4. The micromachined gyroscope of claim3, wherein the driving means are parallel plate electrostatic actuators.5. The micromachined gyroscope claim 1, wherein the three masses aresensing masses configured to move when the micromachined gyroscope isrotated.
 6. The micromachined gyroscope of claim 1, further comprising aduplicate of the three-mass configuration, wherein the duplicate isconfigured to oscillate in the first direction but in an opposite phaseof the three-mass configuration.
 7. The micromachined gyroscope of claim1 wherein, the three masses are configured to oscillate linearly.
 8. Amethod for designing a micromachined gyroscope comprising: selecting afirst mass having a first weight m₁; selecting a second mass having asecond weight m₂; selecting a third mass having a third weight m₃;selecting a first spring constant k₁ for a mechanical connection betweenthe first mass and a substrate; selecting a second spring constant k₂for a mechanical connection between the second mass and the substrate;selecting a third spring constant k₃ for a mechanical connection betweenthe third mass and the substrate; selecting a fourth spring constant k₁₂for a mechanical connection between the first mass and the second mass;and selecting a fifth spring constant k₂₃ for a mechanical connectionbetween the second mass and the third mass, wherein:[(k₁+k₁₂)/m₁]˜[k₃+k₂₃)/m₃]; and wherein, during operation of themicromachined gyroscope:[(k ₂ +k ₁₂ +k ₂₃)/m ₂]>>[(k ₁+k₁₂)/m ₁ ]>[k ₃ +k ₂₃)/m ₃].
 9. A methodaccording to claim 8, further comprising: selecting a mechanicalamplification between a movement of the first mass and a movement of thethird mass; and dimensioning the second spring constant k₂ in view ofthe mechanical amplification.